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Abstract:We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $\omega=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $\omega=3$ analogue. Counterintuitively the problem hardens as $\omega$ decreases -- the induction's feed thins -- so $\omega=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\le\beta(N)$, proved for all $N\ge10$ by Olson's addition theorem and elementary Chebyshev bounds. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\min\{B_{N_b}/6,\,1/5\}$ is $\omega=2$ representable, unconditionally. As an application we give the first complete proof of the rational $\omega=3$ statement -- every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions -- that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $\omega\ge3$. What remains open is the $\omega=2$ regime below this threshold, which we reduce to a single explicit conjecture -- that the gap-free floor of a semiprime subset-sum set tends to zero.

Submission history

From: Shisheng Li [view email]
[v1] Sat, 13 Jun 2026 07:10:39 UTC (174 KB)